If the lines whose equations are `y=m_1x+c_1,y=m_2x+c_2and y=m_3x+c_3` are concurrent, then show that `m_1(c_2-c_3)+m_2(c_3-c_1)+m_3(c_1-c_2)=0`.

If three lines whose equations are y = m_1x + c_1 , y = m_2x + c_2 and y = m_3x + c_3 are concurrent, then show that m_1(c_2 - c_3) + m_2 (c_3 -c_1) + m_3 (c_1 - c_2) = 0 .

Show that the area of the triangle formed by the lines whose equations are : y=m_1x+c_1, y= m_2 x+c_2 and x=0 is : ((c_1-c_2)^2)/(2|m_1-m_2|) .

A triangle has its three sides equal to a,b and c. If the co-ordinates of its vertices are A(x_1y_1),B(x_2,y_2) and C(x_3,y_3) , show that {:|(x_1,y_1,2),(x_2,y_2,2),(x_3,y_3,2)|^2 = (a+b+c)(b+c-a)(c+a-b)(a+b-c)

If the points A(x_1,y_1), B(x_2,y_2) and C(x_3,y_3) are collinear, then area of triangle ABC is:

If y = e^(m tan^-1 x) , then show that (1 +x^2) y_2 + (2x - m) y_1= 0 .

The pair of linear equations a_1x+b_1y+c_1=0' and a_2x+b_2y+c_2=0' have a unique solution, if _____ .

If equations ax^(2)+bx+c=0 (where a,b,c epsilonR and a!=0 ) and x^(2)+2x+3=0 have a common root, then show that a:b:c=1:2:3

If the straight lines a x + m ay + 1 =0 b x + ( m +1) by + 1 =0 cx + (m+2) cy +1 =0, m ne0 are concurrent, then a, b, c are in :

Prove that (-a,-a/2) is the orthocentre of the triangle formed by the lines y = m_(i) x + a/(m_(i)) , I = 1,2,3, m_(1) m_(2) m_(3) being the roots of the equation x^(3) - 3x^(2) + 2 = 0

Find the order of the family of curves y=(c_1+c_2)e^x+c_3e^(x+c_4)